Remark 2.5 The notion of \(n\) -metric on Lie groupoids for \(n\ge 3\)  was introduced in [1]}. This is just a Riemannian metric on the set of \(n\) -composable arrows \(G^{(n)}\)  that is invariant by the canonical \(S_{n+1}\) -action on \(G^{(n)}\)  and transverse to one (hence to all) face map \(G^{(n)}\rightarrow G^{(n-1)}\) . We can push this \(n\) -metric forward with the different face maps \(G^{(n)}\rightarrow G^{(n-1)}\)  to define an \((n-1)\) -metric on \(G^{(n-1)}\)  in such a way these face maps become Riemannian submersions. One can use this process to obtain \(r\) -metrics \(\eta ^{(r)}\)  on \(G^{(r)}\)  for all \(0\le r\le n-1\)  so that we get Riemannian submersions \((G^{(r)},\eta ^{(r)})\rightarrow (G^{(r-1)},\eta ^{(r-1)})\) .

